derivative of (x^2-2x-3/((x-1)^2))

解

\frac{d}{d x} (\frac{x ^{2} - 2 x - 3}{( x - 1 ) ^{2}})

\frac{8}{( x - 1 ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} - 2 x - 3}{( x - 1 ) ^{2}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x ^{2} - 2 x - 3 ) ( x - 1 ) ^{2} - \frac{d}{d x} ( ( x - 1 ) ^{2} ) ( x ^{2} - 2 x - 3 )}{( ( x - 1 ) ^{2} ) ^{2}}

\frac{d}{d x} (x^{2} - 2 x - 3) = 2 x - 2

\frac{d}{d x} ((x - 1)^{2}) = 2 (x - 1)

= \frac{( 2 x - 2 ) ( x - 1 ) ^{2} - 2 ( x - 1 ) ( x ^{2} - 2 x - 3 )}{( ( x - 1 ) ^{2} ) ^{2}}

簡素化

\frac{( 2 x - 2 ) ( x - 1 ) ^{2} - 2 ( x - 1 ) ( x ^{2} - 2 x - 3 )}{( ( x - 1 ) ^{2} ) ^{2}} : \frac{8}{( x - 1 ) ^{3}}

= \frac{8}{( x - 1 ) ^{3}}